


Bi |wii|p*s1 dx )p/p*s1
crs1prNBiBi |wi(x)i(x)-wi(y)i(y)|p|x-y|N+ps1dxdy+ 



Bi |wii|pdx.

The first term on the right hand side is estimated by means of the following Caccioppoli type inequality (see [1]})
\(& \int _{B_i}\int _{B_i} \frac{|\tilde{w}_i(x)\psi _i(x)-\tilde{w}_i(y)\psi _i(y)|^p}{|x-y|^{N+ps_1}}dxdy \nonumber \\&\le C \sum _{(\ell ,s)} \int _{B_i}\int _{B_i} \frac{|\psi _i(x)-\psi _i(y)|^\ell }{|x-y|^{N+s\ell }}(\tilde{w}_i(x)^\ell +\tilde{w}_i(y)^\ell )dxdy \nonumber \\&\quad + C \sum _{(\ell ,s)} \Big (\sup _{y\in {\rm supp}(\psi _i)} \int _{\mathbb {R}^N\setminus B_i} \frac{\tilde{w}_i(x)^{\ell -1}dx}{|x-y|^{N+s\ell }}\Big ) \int _{B_i}\tilde{w}_i\psi _i^p+ \int _{\Omega } |f| \tilde{w}_i\psi _i^p,\) 
